Two Years Experience in Teaching Mathematics for Engineers with Wolfram Alpha, Study notes of Mathematics

Download Two Years Experience in Teaching Mathematics for Engineers with Wolfram Alpha and more Study notes Mathematics in PDF only on Docsity! Experience with Teaching Mathematics for Engineers with the Aid of Wolfram Alpha Petr Nečesal and Jan Pospı́šil Abstract—Recent development of answer engines such as Wolfram Alpha brought a huge computational power to the end devices such as mobile phones and as a consequence students now can perform many calculus tasks very easily. Teachers could either forbid using such tools and insist on often quite long time consuming manual calculations or they can incorporate their usage in the education process that obviously requires update of syllabuses and teaching materials. In this paper we present two years experience with teaching mathematics for engineers with the aid of Wolfram Alpha that students can use not only in classes, but also during the written part of examination. Index Terms—mathematics, calculus, Wolfram Alpha, multi- ple choice tests, generator for written exams, content manage- ment system. I. INTRODUCTION STOP teaching calculating, start teaching math - this is a marketing phrase used by Wolfram Research to support their products Mathematica and Wolfram Alpha. Though be- ing rather tentative it surprisingly opened a vivid discussion among academics. From a black and white perspectives there exist two groups of mathematics teachers, ones insisting on manual calculations being the only correct way how to train the brain and learn calculus properly, and the other group claiming that it is a waste of time doing time consuming calculations manually when a computer can give the answer almost immediately. Providing the list of all arguments of both groups goes beyond the scope of this paper. Choosing a gray position somewhere in between we rather present a practical information based on two years experience of using Wolfram Alpha as a practical aid that engineering students are allowed to use not only in classes, but also during the written part of examination (examinations of almost all courses have written and oral part). We introduce Wolfram Alpha and its usefulness in the mathematics courses for engineers. Together with the system TRIAL that during a decade evolved from an online database of mathematics problems to a living web portal playing a role of learning management system for mathematics education. In the last section we list our experiences with practical usage of Wolfram Alpha by students and teachers, we show the most common types of errors students do, we list some problems that cannot be solved by Wolfram Alpha (at least in its current version) and provide some hints and advices for educators. Manuscript received July 17, 2012; revised August 3, 2012. This work was partially supported by the Czech Ministry of Education under projects OP VK CZ.1.07/2.2.00/15.0377 and OP VK CZ.1.07/2.4.00/17.0100. Both authors are with the Department of Mathematics, University of West Bohemia, Plzeň, Czech Republic. E-mail: pnecesal@kma.zcu.cz, honik@kma.zcu.cz. II. WOLFRAM ALPHA Wolfram Alpha is an answer engine product created and developed by Wolfram Research. It is based on Mathematica and works as an online service that answers queries directly by computing the answers rather than providing a list of web pages where the answer can be found. It was released in May 2009 and currently it includes hundreds of datasets from many disciplines (historical, financial, economic, ge- ographical, meteorological, lexical, and many more). From the mathematics point of view it encompasses computer algebra, symbolic and numerical computation, visualization, and statistics capabilities. Public part of Wolfram Alpha is available at the web address www.wolframalpha.com. Wolfram Alpha brings the biggest added value to Math- ematica, namely the free form input1. It is capable of responding to particularly phrased natural-language questions such as plot xˆ2-yˆ2 or more complex questions such as find saddle points of xˆ2-yˆ2. It displays its ”Input interpretation” of such a question using standardized phrases, e.g. saddle points | xˆ2 - yˆ2, or one can provide the standard Mathematica language input, e.g. Plot[xˆ2-yˆ2, {x; -3; 3}, {y; -3; 3 }]. More details about Wolfram Alpha can be found directly at the web server including many examples. Here we mention some advantages of using it at the mathematics courses for engineers. At first it is capable of almost all calculus tasks students need, it finds the limits, it can differentiate and in- tegrate, solve algebraic systems of equations, find properties of matrices, solve some simple differential equations, find the series expansions, and almost everything can be solved both symbolically and numerically. Secondly, it is free and when online then it is available not only on computers with a web browser, but also on all modern end devices including mobile phones and tablets. Well, Wolfram Research recently released a Wolfram Alpha Pro paid subscription with more features that also led to an increase in advertisement on the free site, but the free version still covers all the tasks that students might require in the mathematics courses. One of the disadvantage might be the online access requirement. During its three years history, Wolfram Alpha proved to be a reliable system allowing more than 500 students in the examination rooms to use it at the same time often without the need of an extra computational time. Requirement of the Internet access now therefore belongs to the category of blackout threats. Other advantages and disadvantages with respect to math- ematics teaching will be discussed below. 1The newest Mathematica now also offers the Wolfram Alpha input. Proceedings of the World Congress on Engineering and Computer Science 2012 Vol I WCECS 2012, October 24-26, 2012, San Francisco, USA ISBN: 978-988-19251-6-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCECS 2012 III. WEB-BASED SYSTEM TRIAL A system called TRIAL was created and is continu- ously being developed at the Department of Mathematics (www.kma.zcu.cz) at the Faculty of Applied Sciences at the University of West Bohemia in Plzeň, Czech Republic. The history of TRIAL goes back to year 2002 and its original purpose was to create an online database of mathematics problems for the first year students at the technical faculties. Problems were then used in written examination tests and students could practice the trial versions (here the name TRIAL comes from) of the examination. Since then the database has grown up to several hundreds types of problems (each problem is available in many random mutations) and TRIAL is used in classes and examinations of ten mayor courses at five faculties for roughly 3500 students every year. The public part of TRIAL is available at the web address trial.kma.zcu.cz. Thanks to its size and content the server became one of the most visited web server at the university with the overall access rate greater than 1.2 millions individual visits per year. Nowadays TRIAL offers the following services: 1) generating of written tests (problem, solution, results), usually number of students = number of different written tests, 2) displaying results of written tests, 3) generating lecture notes for students (definitions, the- orems, proofs, etc.), 4) generating lecture notes for lecturers and assistants, 5) discussion forum. More details about TRIAL can be found in [1]. Here we mention two major advantages of using TRIAL together with Wolfram Alpha. The first is the uniqueness of problems in each student’s test. None of the tests is the same, only the difficulty is consistent. This fact forces students to really work alone, individually formulate the questions for the answer engine and correctly get the results. A second advantage comes into account especially before the tests - a discussion forum. It is a place where students discuss with teachers and among themselves problems and their solution, they share the knowledge of questioning the answer engine and surprisingly they ask and discuss problems engineering students never asked before going for example deep into mathematical analysis. Having more than a hundred forum threads every term, mathematics became one of the most discussed topics for first year students. Exercises in TRIAL are of the following types: 1) standard exercises, students has to solve given prob- lems and provide step-by-step solution, see for example Exercises 1 and 2 at the end of this section. 2) multiple choice tests, students only mark correct an- swers, see for example Exercises 3 and 4, 3) fill-in tests, students have to calculate given problems (usually very easy) and provide only the result, they have to fill-in a missing information in the text or sketch a graph, see Exercises 5, 6 and 7. Let us close this section by some exercises, which were automatically generated by the system TRIAL and that are regularly included in the written tests in the first term. Exercise 1. Let {an} be a real sequence and let a ∈ N be Fig. 1. One sample problem (in Czech language) generated from the system TRIAL together with its step-by-step solution. f(x, y) = x2y2 x4+y4 f(x, y) = √ |xy| f(x, y) = sinx+ sin y f(x, y) = sin(xy) (x2 + y2 + z2)2 = x2 + y2 − z2 (2− √ x2 + y2)2 = 1− z2 Fig. 2. System TRIAL contains also a lot of addons, for example graphs of functions of two variables and implicitly given surfaces for anaglyph 3D glasses. Proceedings of the World Congress on Engineering and Computer Science 2012 Vol I WCECS 2012, October 24-26, 2012, San Francisco, USA ISBN: 978-988-19251-6-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCECS 2012